Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

9. Let a, b, and d be positive integers. Prove that if (a, b) = 1 and d divides a, then (d, b) = 1.

Prove that AB=An(-B) = (–B) − (− A) = −[(-A) U B]. Also, give three expressions for An B and AUB, in terms of complements.

Let n be an integer, use the definition of even to prove that 10n+ 4 is even. 10n+4=0x0 Since is a is an integer.is is also an integer. Since 10n+4 equals 2 times an integer, 10n+ 4 is even.

Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

Let a, b be positive integers, and let p be a prime integer. Let the notation n|m means m divides n with no remainder. Show: (p|(a² + b²))) → plb.

Let a,b,c,d, and e be integers with a not equal to 0. if (ab,ac)=e and (b,c)=d, then e divides ad. Prove

Let a and b be integers, and let d be a positive common multiple of a and b. Show that the following two statements are equivalent.  (1) d <= c for every positive common multiple c of a and b. (2) d divides c for every common multiple c of a and b.

Let a, b, and c, be integers. Suppose that c ≄ 0. Prove that a|b if and only if ac|ab.

Let a, b, and c be integers. Prove that if a divides b and a divides c,then a divides (b-c).